Re: Epistemology 201: The Science of Science
From: robert j. kolker (nowhere_at_nowhere.net)
Date: 03/20/05
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Date: Sun, 20 Mar 2005 12:31:09 -0500
Lester Zick wrote:
> On Sun, 20 Mar 2005 11:26:04 -0500, "robert j. kolker"
> <nowhere@nowhere.net> in comp.ai.philosophy wrote:
>
>
>>Lester Zick wrote:
>>
>>>There is a distinction between mathematics, Bob, and modern math.
>>
>>So you claim. Now show the difference. Be very explicit.
>
>
> The difference between your first modern math definition for a circle
> which actually defines a sphere and your second Euclidean definition
You said there was a difference between mathematics and modern math.
That is a very general statement. Substantiate it. Address yourself to
the question I asked.
> for a circle which allows you to pretend that circles are well defined
> as the set of all points equidistant from any point without definition
> for spatial dimensionality that allows you to pretend dimensionality
Schmuck. What is the dimension of a plane, in the sense of maximal
number of mutually orthagonal lines lying on a plane? Think! Put all 13
of your neurons to work.
> is just so much vulcanized rubber.
When I made the defnition complete (my appologies for the initial
omission) I made it plain that it was a figure on a plane. Now what can
be plainer than a plane. Dimnsionality for vector spaces is not rubber,
it is the cardinality of the maximal set of linearly independent vectors
in the vector space. One must show that all maximal linearly independent
sets have ths same cardinality (easy for finite dimensional vector
spaces, not so easy for infinite dimensional vector spaces).
Except for specifying that a circle is a figure on a plane the
definition gives no refernce to dimensionality whatsovery. In fact one
can show a circle on a plane (a two dimensional surface) is a one
dimension set since it can be parametrically generated by one variable,
to with the angle an arbitrary radius makes with a reference radius.
That conclusion is not immediately clear from the definition. A lot of
theorems have to proven to show that.
Bob Kolker
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